The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils

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(1)Int. J. Appl. Math. Comput. Sci., 2013, Vol. 23, No. 1, 29–33 DOI: 10.2478/amcs-2013-0003. APPLICATION OF THE DRAZIN INVERSE TO THE ANALYSIS OF DESCRIPTOR FRACTIONAL DISCRETE–TIME LINEAR SYSTEMS WITH REGULAR PENCILS TADEUSZ KACZOREK Faculty of Electrical Engineering Białystok Technical University, ul. Wiejska 45D, 15-351 Białystok, Poland e-mail: kaczorek@isep.pw.edu.pl. The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example. Keywords: Drazin inverse, descriptor, fractional system, discrete-time system, linear system.. Dedicated to Bogumiła Adamczyk on the occasion of her birthday. 1. Introduction Descriptor (singular) linear systems were considered in many papers and books (Bru et al., 2000; 2002; 2003; Campbell et al., 1976; Dai, 1989; Dodig and Stosic, 2009; Fahmy and O’Reill, 1989; Kaczorek, 1992; 2004; 2007b; 2011a; 2011d; Van Dooren, 1979; Virnik, 2008). The eigenvalues and invariants assignment by state and output feedbacks were investigated by Fahmy and O’Reill (1989), Gantmacher (1960), Kaczorek (2004) as well as Kucera and Zagalak (1988), and the realization problem for singular positive continuous-time systems with delays was examined by Kaczorek (2007b). The computation of Kronecker’s canonical form of the singular pencil was analyzed by Van Dooren (1979). Positive linear systems with different fractional orders were addressed by Kaczorek (2010), who also discussed selected problems of fractional linear systems theory (Kaczorek, 2011b). A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of the state of the art in positive systems theory is given by Farina and Rinaldi (2000), Commalut and Marchand (2006), as well as Kaczorek (2002; 2007a). A variety of models having positive behavior can be found. in engineering, economics, social sciences, biology and medicine, etc. Descriptor standard positive linear systems using the Drazin inverse were addressed by Bru et al. (2003; 2000; 2002) and Kaczorek (2002; 2011b). The shuffle algorithm was applied to check the positivity of descriptor linear systems by Kaczorek (2011a), while the stability of positive descriptor systems was investigated by Virnik (2008). Reduction and decomposition of descriptor fractional discrete-time linear systems were considered by Kaczorek (2011d), who also introduced a new class of descriptor fractional linear discrete-time systems (Kaczorek, 2011c). In this paper the Drazin inverse of matrices will be applied to find the solutions of the state equations of descriptor fractional discrete-time linear systems with regular pencils. The paper is organized as follows. In Section 2 the state equation of the descriptor fractional linear discrete-time system and some basic definitions of the Drazin inverse are recalled. The solution to the state equation is given in Section 3. The proposed method is illustrated by numerical examples in Section 4. Concluding remarks are given in Section 5. The following notation will be used: R is the set of real numbers, Rn×m is the set of n × m real matrices and Rn = Rn×1 , Z+ is the set of nonnegative integers, N is the set of natural numbers, In is the n × n identity matrix, ker A is the kernel of the matrix, C is the field of complex numbers..

(2) T. Kaczorek. 30. 2. Preliminaries Consider the descriptor fractional discrete-time linear system EΔα xi+1 = Axi +Bui ,. i ∈ Z+ ,. det[Ez − A] = 0 for some z ∈ C.. (2). The fractional difference of the order α is defined by (Kaczorek, 2011b) Δ xi =. i . ck xi−k ,. n − 1 < α < n ∈ N,. (3). k=0. where. α ck = (−1) , k k. and α k =. k = 0, 1, . . . ,. 1 α(α − 1) . . . (α − k + 1) k!. (4a). for k = 0, for k = 1, 2, . . .. (4b). Exi+1 = F xi −. Eck+1 xi−k + Bui ,. (9a) (9b) (9c). ¯ defined by (8). where q is the index of E ¯ D of a square matrix E ¯ always The Drazin inverse E exists and is unique (Campbell et al., 1976; Kaczorek, ¯ = 0 then E¯ D = E¯ −1 . Some methods for 1992). If det E computation of the Drazin inverse are given by Kaczorek (1992). Lemma 1. (Campbell et al., 1976; Kaczorek, 1992) The matrices E¯ and F¯ defined by (7b) satisfy the following equalities: ¯=E ¯ F¯ D , ¯ F¯ , F¯ D E F¯ E¯ = E ¯ D , F¯ D E ¯D = E ¯ D F¯ D , E¯ D F¯ = F¯ E ¯ = {0}, ker F¯1 ∩ ker E J 0 ¯ E=T T −1 , 0 N A1 0 ¯ F =T T −1 , 0 A2 −1 J 0 D ¯ E =T T −1 , 0 0. (10a) (10b). (10c). det T = 0, J ∈ Rn1 ×n1 is nonsingular,. Substitution of (3) into (1) yields i . ¯ ¯E ¯D = E ¯ D E, E ¯DE ¯E ¯D = E ¯D, E ¯DE ¯ q+1 = E ¯q, E. 0 < α < 1, (1). where α is the fractional order, xi ∈ Rn is the state vector, ui ∈ Rm is the input vector and E, A ∈ Rn×n , B ∈ Rn×m . It is assumed that det E = 0, but the pencil (E, A) is regular, i.e.,. α. Definition 2. (Kaczorek, 1992) A matrix E¯ D is called the ¯ ∈ Rn×n if it satisfies the conditions Drazin inverse of E. N ∈ Rn2 ×n2 i ∈ Z+ ,. is nilpotent,. A2 ∈ Rn2 ×n2 ,. A1 ∈ Rn1 ×n1 ,. n1 + n2 = n,. k=1. (5a) where F = A − Ec1 .. (10d). (5b). 3. Solution to the state equation. Assuming that det[Ec − F ] = 0. for some c ∈ C,. (6). and premultiplying (5a) by [Ec − F ]−1 , we obtain ¯ i+1 = F¯ xi − Ex. ¯E ¯ D )F¯ F¯ D = In − E¯ E¯ D (In − E and (In − E¯ E¯ D )(E¯ F¯ D )q = 0.. i . ¯ k+1 xi−k + Bu ¯ i, Ec. (7a). k=1. Theorem 1. The solution to Eqn. (7a) with an admissible initial condition x0 is given by ¯ D Ex ¯ 0 ¯ D F¯ )i E xi = (E. where E¯ = [Ec − F ]−1 E, ¯ = [Ec − F ]−1 B. B. In this section the solution to the state equation (1) will be presented by the use of the Drazin inverses of the matrices E¯ and F¯ .. F¯ = [Ec − F ]−1 F, (7b). Definition 1. (Kaczorek, 1992) The smallest nonnegative integer q satisfying ¯ q+1 ¯ q = rank E rank E ¯ ∈ Rn×n . is called the index of the matrix E. +. i−1 . k . ¯ D (E¯ D F¯ )i−k−1 Bu ¯ j+1 xk−j ¯ k− E Ec j=1. k=0. ¯ D − In ) + (E¯ E. q−1 . ¯ i+k , ¯ F¯ D )k F¯ D Bu (E. k=0. (11). (8) ¯ where q is the index of E..

(3) Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems. . . Proof. Using (11) and taking into account (9) and (10), we obtain ¯ i+1 Ex ¯ 0 ¯ E ¯ D F¯ )i+1 E¯ D Ex = E( +. i . k . D ¯ D ¯ i−k ¯ ¯ ¯ ¯ j+1 xk−j+1 E E (E F ) Ec Buk − j=1. k=0. ¯ E ¯E ¯ D − In ) + E(. q−1 . and admissible initial conditions for given input ui , i ∈ Z+ . The pencil of (16) is regular since. z 0. = 2z, det[Ez − A] =. −1 2. α 0 F = A − Ec1 = A + Eα = , 1 −2 q = 1. (17). ¯ i+k+1 (E¯ F¯ D )k F¯ D Bu. k=0. For c = 1 the matrices (7b) have the forms. ¯ D Ex ¯ 0 ¯ D F¯ )i E = F¯ (E i−1 k . D ¯ i−k ¯ ¯ j+1 xk−j + Bu ¯ i ¯ Ec Buk − + (E F ). −1 1−α 0 1 E¯ = [Ec − F ]−1 E = −1 2 0 1 2 0 2 0 = = , 1 0 2(1 − α) 1 0. j=1. k=0. ¯E ¯D) + (In − E. q−1 . ¯ k F¯ F¯ D Bu ¯ i+k (−F¯ D E). k=0. (12) and F¯ xi ¯D. i. ¯D. ¯ 0 = F¯ (E F¯ ) E Ex +. i−1 . k . ¯ k− ¯ D (E ¯ j+1 xk−j ¯ D F¯ )i−k−1 Bu F¯ E Ec j=1. k=0. ¯E ¯ D − In ) + F¯ (E. q−1 . 31. ¯ i+k . (E¯ F¯ D )k F¯ D Bu. −1 1−α 0 −1 ¯ F = [Ec − F ] F = −1 2 1 2α 0 = = 1 −2(1 + α) 2(1 − α) ¯ = [Ec − F ]−1 B = 1 − α B −1 1 2 = = 2(1 − α) 3 − 2α. −1 . 0 2. . 2 2. 0 0. α 1. 0 −2. 1 2. ¯ = T −1 E. . (14). k=1. . .. 2 0 0 0. . T,. T =. 1 0 1/2 1. (19). and. Thus, the solution (11) satisfies Eqn. (7a).. . ¯ D = T −1 E. From (11), for i = 0 we have ¯ 0 +(E ¯ D Ex ¯E ¯ D −In ) x0 = E. ,. Using (10c) and (18), we obtain. Hence ¯ k+1 xi−k = Bu ¯ i. Ec. . (18) (13). ¯ i+1 − F¯ xi − Ex. . −1 0 1 −1. k=0. j . . q−1 . ¯ k . (15) ¯ F¯ D )k F¯ D Bu (E. 4. Example Find the solution xi to Eqn. (1) with α = 0.5 and the matrices 1 0 0 0 1 E= , A= , B= , 0 0 1 −2 2 (16). 1/2 0 0 0. . T =. 1/2 0 −1/4 0. .. (20). Note that. 1 det F¯ =. 1. k=0. The set of admissible initial conditions x0 for given input ui is given by (15). In a practical case, for ui = ¯ 0 . Thus, the equation ¯ D Ex 0, i ∈ Z+ we have x0 = E ¯ i+1 = Axi has a unique solution if and only if x0 ∈ Ex ¯E ¯ D , where ‘Im’ denotes the image. ImE. . 0. = −1 = 0 −1. and F¯ D = F¯ −1 =. . 1 1. 0 −1. (21). .. (22). Taking into account that . 1/2 0 1 0 1/2 0 = , −1/4 0 1 −1 −1/4 0 2 0 1/2 0 1 0 = = 1 0 −1/4 0 1/2 0 (23). ¯ D F¯ = E ¯E ¯D E.

(4) T. Kaczorek. 32 and using (11), we obtain i 1/2 0 1 0 xi = x0 −1/4 0 −1/2 0 i−k−1 i−1 1/2 0 1/2 0 + −1/4 0 −1/4 0 k=0 ⎧ ⎫ (24) k ⎨ ⎬ 2 0 2 × cj+1 xk−j ] uk − 1 0 ⎩ 2 ⎭ j=1 0 0 1 + ui , 1/2 −1 −1 where the coefficients cj are defined by (4a) for α = 0.5. From (24), for i = 0 we have 1 0 0 x0 = x0 + u0 . (25) 1/2 0 −2. Bru, R., Coll, C. and Sanchez, E. (2002). Structural properties of positive linear time-invariant difference-algebraic equations, Linear Algebra and Applications 349(1–3): 1–10. Campbell, S.L., Meyer, C.D. and Rose, N.J. (1976). Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM Journal on Applied Mathematics 31(3): 411–425. Commalut, C. and Marchand, N. (2006). Positive Systems, Lecture Notes in Control and Information Sciences, Vol. 341, Springer-Verlag, Berlin. Dai, L. (1989). Singular Control Systems, Lectures Notes in Control and Information Sciences, Vol. 118, Springer-Verlag, Berlin. Dodig, M. and Stosic, M. (2009). Singular systems state feedbacks problems, Linear Algebra and Its Applications 431(8): 1267–1292.. Hence, for given u0 , the admissible initial condition x0 should satisfy (25).. Fahmy, M.H, and O’Reill, J. (1989). Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalues assignment, International Journal of Control 49(4): 1421–1431.. 5. Concluding remarks. Farina, L. and Rinaldi, S. (2000). Positive Linear Systems, J. Willey, New York, NY.. The Drazin inverse of matrices has been applied to find the solutions of the state equations of the descriptor fractional discrete-time systems with regular pencils. The equality (15) defining the set of admissible initial conditions for given inputs has been derived. The proposed method has been illustrated by a numerical example. Comparing the presented method with that based on the Weierstrass decomposition of the regular pencil (Kaczorek, 2011c), we may conclude that the proposed approach is computationally attractive since the Drazin inverse of matrices can be computed by the use of well-known numerical methods (Kaczorek, 1992). The presented method can be extended to descriptor fractional continuous-time linear systems. An open problem is the extension of the deliberations to standard and positive continuous-discrete descriptor fractional linear systems.. Acknowledgment This work was supported by the National Science Centre in Poland under the grant no. N N514 6389 40.. References Bru, R. , Coll, C., Romero-Vivo S. and Sanchez, E. (2003). Some problems about structural properties of positive descriptor systems, in L. Benvenuti, A. Santis and L. Farina (Eds.), Positive Systems, Lecture Notes in Control and Information Sciences, Vol. 294, Springer, Berlin, pp. 233–240. Bru, R., Coll, C. and Sanchez, E. (2000). About positively discrete-time singular systems, in N.E. Mastorakis (Ed.) System and Control: Theory and Applications, Electrical and Computer Engineering Series, World Scientific and Engineering Society, Athens, pp. 44–48.. Gantmacher, F.R. (1960). The Theory of Matrices, Chelsea Publishing Co., New York, NY. Kaczorek, T. (1992). Linear Control Systems, Vol. 1, Research Studies Press, J. Wiley, New York, NY. Kaczorek, T. (2002). Positive Springer-Verlag, London.. 1D. and. 2D. Systems,. Kaczorek, T. (2004). Infinite eigenvalue assignment by an output/feedback for singular systems, International Journal of Applied Mathematics and Computer Science 14(1): 19–23. Kaczorek, T. (2007a). Polynomial and Rational Matrices. Applications in Dynamical Systems Theory, Springer-Verlag, London. Kaczorek, T. (2007b). Realization problem for singular positive continuous-time systems with delays, Control and Cybernetics 36(1): 47–57. Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(3): 453–458. Kaczorek, T. (2011a). Checking of the positivity of descriptor linear systems by the use of the shuffle algorithm, Archives of Control Sciences 21(3): 287–298. Kaczorek, T. (2011b). Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin. Kaczorek T. (2011c). Singular fractional discrete-time linear systems, Control and Cybernetics 40(3): 1–8. Kaczorek T. (2011d). Reduction and decomposition of singular fractional discrete-time linear systems, Acta Mechanica et Automatica 5(4): 1–5. Kucera, V. and Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems, Automatica 24(5): 653–658..

(5) Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems. . . Van Dooren, P. (1979). The computation of Kronecker’s canonical form of a singular pencil, Linear Algebra and Its Applications 27: 103–140. Virnik, E. (2008). Stability analysis of positive descriptor systems, Linear Algebra and Its Applications 429(10): 2640–2659. Tadeusz Kaczorek received the M.Sc., Ph.D. and D.Sc. degrees in electrical engineering from the Warsaw University of Technology in 1956, 1962 and 1964, respectively. In the years 1968–69 he was the dean of the Electrical Engineering Faculty, and in the period of 1970–73 he was a deputy rector of the Warsaw University of Technology. In 1971 he became a professor and in 1974 a full professor at the same university. Since 2003 he has been a professor at Białystok Technical University. In 1986 he was elected a corresponding member and in 1996 a full member of the Polish Academy of Sciences. In the years 1988–1991 he was the director of the Research Centre of the Polish Academy of Sciences in Rome. In 2004 he was elected an honorary member of the Hungarian Academy of Sciences. He has been granted honorary doctorates by nine universities. His research interests cover systems theory, especially singular multidimensional systems, positive multidimensional systems, singular positive 1D and 2D systems, as well as positive fractional 1D and 2D systems. He initiated research in the field of singular 2D, positive 2D and positive fractional linear systems. He has published 24 books (six in English) and over 1000 scientific papers. He has also supervised 69 Ph.D. theses. He is the editor-in-chief of the Bulletin of the Polish Academy of Sciences: Technical Sciences and a member of editorial boards of ten international journals.. Received: 25 April 2012 Revised: 24 October 2012. 33.

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